Abstract
Graphs and Algorithms We conjecture Ore and Erdős type criteria for a balanced bipartite graph of order 2n to contain a long cycle C(2n-2k), where 0 <= k < n/2. For k = 0, these are the classical hamiltonicity criteria of Moon and Moser. The main two results of the paper assert that our conjectures hold for k = 1 as well.
Highlights
One of the classical problems of graph theory is the study of sufficient conditions for a graph to contain a Hamilton cycle
If dG(x) + dG(y) ≥ n − k + 1 for every pair of non-adjacent x ∈ X and y ∈ Y, G contains a cycle of length 2n − 2k
Given a balanced bipartite graph G = (X, Y ; E), one defines a k-biclosure BClk(G) of G as the graph obtained from G by succesively joining pairs of non-adjacent vertices x ∈ X and y ∈ Y, with degree sum of at least k, until no such pair remains
Summary
One of the classical problems of graph theory is the study of sufficient conditions for a graph to contain a Hamilton cycle. It follows immediately from Ore’s theorem that the minimal size of a graph of order n ≥ 3 that guarantees hamiltonicity is n−1 2. Let G be a bipartite graph of order 2n, with colour classes X and Y , |X| = |Y | = n ≥ 2, and minimal degree δ(G) ≥ r, 1 ≤ r ≤ n/2. Woodall [14, Thm. 11] gives a complete list of Erdos type conditions for a graph of order n to contain a cycle of length n. By combining Theorems A and B, we obtain a complete Erdos type characterisation of balanced bipartite graphs that do not contain cycles of length 2n − 2 (Theorem 3.6). The last two sections are devoted to proofs of the two main results
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